Collaborative Research: Dynamics, singularities, and variational structure in models of fluids and clustering

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The mathematical structure of numerous important models of dynamic behavior in science and data analysis is fundamentally related with optimality. Fluid motions optimize kinetic energy over time. Many systems in physical and information sciences tend to maximize entropy. Deep learning algorithms are trained by optimizing parameters for clustering and classifying big data sets. This project will improve our mathematical understanding of optimality principles and dynamics in several models of substantial current interest to researchers in a number of disciplines. These range from fluid dynamics and network routing to statistical sampling and data science to aerosol physics and animal ecology. Optimal transport theory will be used in a novel way to model fluid mixture dynamics and understand how fluid surface singularities can form. Gradient descent techniques will be investigated to analyze and improve the convergence of high-dimensional statistical sampling and wave-shape computations. Novel dynamical phenomena in merging-splitting models of clustering will be sought in models relevant to aerosol particle growth in atmospheric dynamics and the sharing of information in financial markets. These investigations will stimulate young researchers and students to participate, and will lead to results to be disseminated at conferences, research institutes, seminars, and lecture series.In particular, this project's research will focus on bringing ideas from variational analysis to bear upon several specific topics of current interest: (1) modeling how incompressible fluids may optimally mix through an entropy-regularized multi-marginal optimal transport formulation, which ought to make numerical computations feasible and may enable a precise characterization of optimal dynamic pathways; (2) demonstrating the formation of singularities on the surface of incompressible fluids in a scenario involving expansion from a corner, through a novel perturbation analysis of a simple geodesic flow; (3) establishing convergence of gradient-like flows to explain coherent-state formation and improve statistical sampling, by developing the use of Lojasiewicz estimates in infinite-dimensional nonlocal models; (4) identifying metastable states and nontrivial temporal dynamics in kinetic models of aggregation and breakup that lack a detailed-balance structure that would drive the syst em to equilibrium.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.